A planar structure of length L and width W is made of two different optical media of refractive indices n 1 = 1.5 and n 2 = 1.44 as shown in figure. If L W , a ray entering from end A B will emerg

Question

A planar structure of length L and width W is made of two different optical media of refractive indices n 1 = 1.5 and n 2 = 1.44 as shown in figure. If L > > W , a ray entering from end A B will emerge from end C D only if the total internal reflection condition is met inside the structure. For L = 9.6 m , if the incident angle θ is varied, the maximum time taken by a ray to exit the plane C D is t × 10 - 9 s , where t is ____. [Speed of light c = 3 × 10 8 m / s ]

TestHub · Accepted Answer

For maximum time taken by a rat to exit, it must travel maximum distance.
So, for this, it should be incident at minimum possible angle i.e., $θ_{C}$ at $n_{1} - n_{2}$ boundary.
Applying Snells' law at $n_{1} - n_{2}$ boundary
$\Rightarrow 1.5 s i n θ_{C} = 1.44 s i n 90 °$
$\Rightarrow s i n θ_{C} = \frac{1.44}{1.50} = \frac{24}{25}$
$∴ s i n θ_{C} = \frac{x}{d} = \frac{24}{25}$
$\Rightarrow d = \frac{25 x}{24}$
$∴$ total length travel by light $= \frac{25}{25} \times 9.6 = 10 m$ $(a s, x_{m a x} = L = 9.6 m)$
$∴ t = \frac{S}{(\frac{C}{n_{1}})} = \frac{10}{\frac{3 \times 10^{n}}{1.5}} (t = \frac{distance travelled}{speed of light in medium})$
$= \frac{1}{2} \times 10^{- 7} = 5 \times 10^{- 9} s e c o n d s$
$t = 50 n s$
$t = 50 \times 10^{- 9}$
$∴ A n s w e r = 50$

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